Infinite-dimensional topology. Prerequisites and introduction.

*(English)*Zbl 0663.57001
North-Holland Mathematical Library, 43. Amsterdam etc.: North-Holland. xii, 401 p. $ 73.25/hbk; Dfl. 150.00/hbk; $ 41.50/pbk; Dfl. 85.00/pbk (1989).

The purpose of writing this book was twofold: (1) to have a textbook for a first-year graduate course on dimension theory, ANR’s and infinite- dimensional topology, and (2) to produce a complete, self-contained exposition of Toruńczyk’s Hilbert cube manifolds characterization theorem. Therefore, the first six chapters (Extension theorems, Elementary plane topology, Elementary combinatorial techniques, Elementary dimension theory, Elementary ANR theory, An introduction to infinite-dimensional topology) serve both purposes (1) and (2), and the last two chapters (Cell-like maps and Q-manifolds, Applications) are the main parts of the book. Furthermore, to make the monograph more readable for potential students, the author has added several exercises, historical notes and a bibliography of 153 items.

Although a survey of Toruńczyk’s work was already written 10 years ago by R. D. Edwards [Sémin. Bourbaki, Vol. 1978/79, Lect. Notes Math. 770, 278-302 (1980; Zbl 0429.57004)] and the necessary background for a student who wanted to study it was available in T. A. Chapman’s lecture notes [Lectures on Hilbert cube manifolds, CBMS Reg. Conf. Ser. Math. 28 (1976; Zbl 0347.57005)] it is nevertheless welcome that we have a monograph on this subject. It could be recommended to anyone who wishes to get familiar with infinite-dimensional topology and at the same time learn about some of its most beautiful results.

Although a survey of Toruńczyk’s work was already written 10 years ago by R. D. Edwards [Sémin. Bourbaki, Vol. 1978/79, Lect. Notes Math. 770, 278-302 (1980; Zbl 0429.57004)] and the necessary background for a student who wanted to study it was available in T. A. Chapman’s lecture notes [Lectures on Hilbert cube manifolds, CBMS Reg. Conf. Ser. Math. 28 (1976; Zbl 0347.57005)] it is nevertheless welcome that we have a monograph on this subject. It could be recommended to anyone who wishes to get familiar with infinite-dimensional topology and at the same time learn about some of its most beautiful results.

Reviewer: D.Repovš

##### MSC:

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

54-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology |

57N20 | Topology of infinite-dimensional manifolds |

54C55 | Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) |

54F45 | Dimension theory in general topology |